What if you had a polynomial expression like in which some of the terms shared a common factor but not all of them? How could you factor this expression? After completing this Concept, you'll be able to factor polynomials like this one by grouping.
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CK-12 Foundation: 0913S Factoring By Grouping
Guidance
Sometimes, we can factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called factor by grouping.
The next example illustrates how this process works.
Example A
Factor .
Solution
There is no factor common to all the terms. However, the first two terms have a common factor of 2 and the last two terms have a common factor of . Factor 2 from the first two terms and factor from the last two terms:
Now we notice that the binomial is common to both terms. We factor the common binomial and get:
Example B
Factor .
Solution
We factor 3x from the first two terms and factor 4 from the last two terms:
Now factor from both terms: .
Now the polynomial is factored completely.
Factor Quadratic Trinomials Where a ≠ 1
Factoring by grouping is a very useful method for factoring quadratic trinomials of the form , where .
A quadratic like this doesn’t factor as , so it’s not as simple as looking for two numbers that multiply to and add up to . Instead, we also have to take into account the coefficient in the first term.
To factor a quadratic polynomial where , we follow these steps:
- We find the product .
- We look for two numbers that multiply to and add up to .
- We rewrite the middle term using the two numbers we just found.
- We factor the expression by grouping.
Let’s apply this method to the following examples.
Example C
Factor the following quadratic trinomials by grouping.
a)
b)
Solution:
Let’s follow the steps outlined above:
a)
Step 1:
Step 2: The number 12 can be written as a product of two numbers in any of these ways:
Step 3: Re-write the middle term: , so the problem becomes:
Step 4: Factor an from the first two terms and a 2 from the last two terms:
Now factor the common binomial :
To check if this is correct we multiply :
The solution checks out.
b)
Step 1:
Step 2: The number 24 can be written as a product of two numbers in any of these ways:
Step 3: Re-write the middle term: , so the problem becomes:
Step 4: Factor by grouping: factor a from the first two terms and a -4 from the last two terms:
Now factor the common binomial :
Watch this video for help with the Examples above.
CK-12 Foundation: Factoring By Grouping
Vocabulary
- It is possible to factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called factoring by grouping .
Guided Practice
Factor by grouping.
Solution:
Let’s follow the steps outlined above:
Step 1:
Step 2: The number 5 can be written as a product of two numbers in any of these ways:
Step 3: Re-write the middle term: , so the problem becomes:
Step 4: Factor by grouping: factor an from the first two terms and from the last two terms:
Now factor the common binomial :
Practice
Factor by grouping.
Factor the following quadratic trinomials by grouping.